ok to clear this up...
original kakashi technique is called Chidori because when at high speed it makes the sound of a thousand birds with is what chidori means.

Raikiri is a more powerful and larger ranged chidori... thus it was renamed raikiri.. As mentioned before it has a different justsu classification because of the difference in power.

chidori original A
raikiri more powerful version S

which although both are elementaly the same one has more power than other.. which basicly makes it a different jutsu. like saying naruto's rasengan is the same as odomarasengan... they are different although both are rasengan just one is larger than the other.
another example... swords... a small sword is call a wakisashi,, a normal is katana and a very large is called great katana. they are all katanas but because they have different length and weight they have different names... so they are not the same